Linear electronic systems are completely characterized by S-parameters. S-parameters are sufficient to predict, based on the superposition principle, the complete response of a linear system to an arbitrary set of stimuli over the frequency range for which the S-parameter data exists. S-parameters characterize the linear relationship b=S*a, where a is the vector the incident waves and b the vector of reflected and transmitted waves. S is the matrix defining the linear relationship between incident and reflected/transmitted waves.
S-parameters are useful for nonlinear systems, but under limited conditions, e.g. a transistor, biased at a fixed set of voltages at the input and output port. If a small RF signal is incident at port 1, linear S-parameter theory and measurement techniques can provide a good approximation to the scattered and reflected waves produced by the transistor. However, S-parameters must be measured for very small signal levels, to make sure the device response is approximately the linear response around the fixed (static) operating point determined by the bias conditions. If the signals used for S-parameter measurements are too large, harmonics are generated by the device that are unaccounted for by S-parameter theory. Beyond the linear response of nonlinear devices at fixed (static) biases, it is often desirable to measure the linear response of nonlinear devices while they are simultaneously stimulated by one or more large signal excitations, such as a large sinusoidal signal at the input, or a multi-carrier communication signal incident on the amplifier. This is the case for power amplifiers when they are actually amplifying a signal. In this case, rigorous mathematical and physical considerations require the linear response of such a system to be given by their X-parameter description:Bpk=Xpk(F)(|A11|)Pk+Xpk,ql(S)(|A11|)Pk−l·Aql+Xpk,ql(T)(|A11|)Pk+l·Aql*  Equation 1Or, equivalently,Bpk=Fpk(|A11|)Pk+Spk,ql(|A11|)Pk−l·Aql+Tpk,ql(|A11|)Pk+l·Aql*  Equation 2Equation 1 shows the contribution to the scattered and transmitted B-waves at port p and harmonic index k given a single large input tone at frequency f, amplitude A11, and an additional small tone at port q and harmonic index, l corresponding to a frequency of 1*f. The objective of the invention is to extract the F, S, and T nonlinear functions for all combinations of port and frequency indices.
Here the nonlinear scattering functions, S and T, depend nonlinearly on the large signals (in this notation a single large input tone, A, incident at port 1), while the dependence on small additional incident waves, a (a vector with components for ports and harmonics or intermods), is linear in both a and conj(a), independently. P=exp(j*phase*t) where phase is the phase of the large-signal incident signal(s). The proper theory of “large-signal S-parameters” we now call X-parameters. The key fact is the requirement of terms linear in both a and conj(a), with different coefficients, the large-signal S-functions and large-signal T-functions, respectively. The present invention is a particularly effective method to implement the measurements at RF, Microwave, and mm-wave and to identify the resulting S and T functions, which are the correct large-signal generalization of linear s-parameters in the case of driven nonlinear systems.
Equation 1 generalizes to the case where there are many large tones. The S and T functions then depend on the amplitudes of each of the large tones (and also the relative phases if some tones are at the same frequency). But the principle of the measurement and identification of the S and T functions from added small tones with orthogonal phases applies directly. Of particular interest is when two large tones are applied at the input of an amplifier, creating energy at the intermodulation products, and for the case of a mixer, two tones at distinct ports (LO and RF) are applied and mixing terms observed at the IF. The invention discussed here applies directly.
One prior art method, e.g. random phase method, is using two standard microwave sources. One source produces a large-signal stimulus while the other source produces small perturbation tones at the fundamental frequency (and harmonics for the augmented model) to probe the linear response. The phase of the probe signal is not controlled, but several phases (multiple measurements) are required so that the (random) phases sample a wide enough range of angles that the X-parameters (S and T functions) can be obtained by regression. At each frequency, this results in an over-determined set of equations for B-waves given a and conj(a) waves at the random phases. These equations are solved by one of a few standard regression analyses. Unfortunately, more measurements than are actually necessary to solve Equation 1 for S and T are required. The regression requires a well-conditioned set of equations from the random measurements. To make sure the equations are well-conditioned, many different phases are required. This takes much more time (proportional to the number of phases). Given the random nature of the multiple measurements, there is always a chance that the resulting equations may still be poorly-conditioned if an unfavorable sample of phases happens to occur. There is a factor of 3 to 6 more measurements taken than optimally required using other methods.
In another prior art method, the offset-frequency-method, small extraction tones are applied simultaneously with the large stimulus signal, to the DUT at frequencies slightly offset from the fundamental of the large signal and also the harmonic or intermod frequencies of the large signal. This allows direct identification of the S and T functions (at each harmonic frequency), by measuring the B-waves at the upper and lower side-bands of the output harmonic spectra. This requires measurements at more frequencies than the present invention (three frequencies in the vicinity of each of the harmonics of interest—including the harmonic frequencies. Another drawback of the offset frequency method is that the magnitude of the frequency difference (offset) between the small a-tones and the large tones must be small for the method to return the S and T functions. This requires the phase reference—a key component of the measurement system—to put out energy at very closely-spaced tones. To get energy at closely spaced tones the phase reference must be pulsed at a low pulse repetition frequency since the tone spacing from the phase reference is directly related to the pulse repetition frequency. The amplitude of the phase reference tones are proportionally related to the duty cycle of the pulse signal. Since the pulse width remains constant and the pulse repetition rate is decreased, to get closely spaced tones the duty cycle decrease linearly with the pulse repetition frequency causing a reduction in the amplitude of the phase reference frequency tones. Since these tones are used to related the phases of the frequencies of interest, any decrease in signal to noise will cause additional measurement uncertainties. The second drawback is that as the offset frequency decreases, phase noise from the measurement system can limit the resolution of the measurements.